Question

If we are given the sets as A=\left\\{ 1,3,5,7,8 \right\\} , B=\left\\{ 3, 5, 8 \right\\} and C=\left\\{ 2,5, 7 \right\\} , then $\left( A-C \right)\times \left( B-C \right)$ is

1. \left\\{ \left( 1,3 \right),\left( 1,8 \right) \right\\}
2. \left\\{ \left( 1,3 \right),\left( 1,8 \right), \left( 3,3 \right), \left( 3,8 \right), \left( 8,3 \right), \left( 8,8 \right) \right\\}
3. \left\\{\left( 3,8 \right), \left( 8,3 \right), \left( 8,8 \right) \right\\}
4. None of these

Answer 1

In this problem we need to calculate the value of $\left( A-C \right)\times \left( B-C \right)$ where A=\left\\{ 1,3,5,7,8 \right\\} , B=\left\\{ 3, 5, 8 \right\\} and C=\left\\{ 2,5,7 \right\\}. We will first calculate the value of $A-C$ by eliminating the common terms of set $A$ and $C$ from set $A$ and write the remaining values in the set $A$. Similarly we can calculate the value of $B-C$ by using the sets $B$ and $C$. After having the values of $A-C$ and $B-C$ we can calculate the value of $\left( A-C \right)\times \left( B-C \right)$ by forming all possible values in the set.

Complete step-by-step solution:
Given sets are A=\left\\{ 1,3,5,7,8 \right\\} , B=\left\\{ 3, 5, 8 \right\\} and C=\left\\{ 2,5,7 \right\\}.
Consider the sets A=\left\\{ 1,3,5,7,8 \right\\} and C=\left\\{ 2,5,7 \right\\}.
In the above both sets we have the common values $5,7$ .
After eliminating the value $5,7$ from set $A$, the set $A$ will be
A=\left\\{ 1,3,8 \right\\}
According to the definition of the operation $A-C$, the value of $A-C$ can be written as
A-C=\left\\{ 1,3,8 \right\\}
Consider the sets B=\left\\{ 3,5,8 \right\\} and C=\left\\{ 2,5,7 \right\\}.
In the above both sets we have the common value $5$ .
After eliminating the value $5$ from set $B$, the set $B$ will be
B=\left\\{ 3,8 \right\\}
According to the definition of the operation $B-C$, the value of $B-C$ can be written as
B-C=\left\\{ 3,8 \right\\}
Now the value of $\left( A-C \right)\times \left( B-C \right)$ can be written as
\left( A-C \right)\times \left( B-C \right)=\left\\{ 1,3,8 \right\\}\times \left\\{ 3,8 \right\\}
Writing all possible combination for the above mentioned relation, then we will have
\left( A-C \right)\times \left( B-C \right)=\left\\{ \left( 1,3 \right),\left( 1,8 \right), \left( 3,3 \right), \left( 3,8 \right), \left( 8,3 \right), \left( 8,8 \right) \right\\}
Hence the option 2 is the correct answer.

Note: The operations on the sets are quite different from the operation on the variables or numerical. In this problem we have to calculate the value of $A-C$, so we have observed red for the common value and eliminated it from the first set which is $A$ and written the remaining elements of the set $A$ as the result of $A-C$. If we have to calculate the value of $C-A$ , then we need to eliminate the common value from set $C$ and write the remaining elements of set $C$ as a result of $C-A$.